Feder-Vardi conjecture, which proposed that every finite-domain Constraint Satisfaction Problem (CSP) is either in P or it is NP-complete, has been solved independently by Bulatov and Zhuk almost ten years ago. Bodirsky-Pinsker conjecture which states a similar dichotomy for countably infinite first-order reducts of finitely bounded homogeneous structures is wide open. In this paper, we prove that CSPs over first-order expansions of finitely bounded homogeneous model-complete cores are either first-order definable (and hence in non-uniform AC$^0$) or L-hard under first-order reduction. It is arguably the most general complexity dichotomy when it comes to the scope of structures within Bodirsky-Pinsker conjecture. Our strategy is that we first give a new proof of Larose-Tesson theorem, which provides a similar dichotomy over finite structures, and then generalize that new proof to infinite structures.

翻译：Feder-Vardi猜想提出每个有限域约束满足问题要么属于P类，要么是NP完全的，该猜想已于约十年前由Bulatov和Zhuk独立解决。Bodirsky-Pinsker猜想则断言有限有界齐性结构的可数无限一阶约化存在类似的二分性，该问题至今仍未解决。本文证明，对于有限有界齐性模型完全核的一阶扩张上的CSP，要么是一阶可定义的（因此属于非均匀AC^0），要么在一阶归约下是L困难的。就Bodirsky-Pinsker猜想所涉及的结构范围而言，这可以说是最一般的复杂度二分性。我们的策略是：首先给出Larose-Tesson定理的新证明（该定理在有限结构上提供了类似的二分性），然后将该新证明推广到无限结构。